arXiv subjects

Michael Grady

Publications and source records attributed to Michael Grady.

At least 19 recordsLinked to original sources

arXiv2023

Possible new phase transition in the 3D Ising Model associated with boundary percolation

In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13\% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent $\nu \sim 1.3$ is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent $\alpha \sim -1.9$. The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different $\nu$ exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there are two different correlation lengths present.

cond-mat.stat-mech↗

arXiv2021

Exploring the 3D Ising gauge-Higgs model in exact Coulomb gauge and with a gauge-invariant substitute for Landau gauge

The Z2 gauge-Higgs model in three dimensions has two different types of phase transition, confinement-deconfinement and Higgs magnetization. Here they are explored through two order parameters, the Coulomb magnetization which is a local order parameter for confinement, and a replica-based gauge-invariant order parameter which tracks the Higgs transition in a way similar to Landau-gauge magnetization. Minimal Coulomb gauge is set exactly, using the polynomial-time minimum-weight matching algorithm of Edmonds. This is a tremendous speed improvement over relaxation/annealing methods and completely eliminates the systematic error. The replica-overlap is also an improvement over relaxation methods for setting Landau gauge, in that it has an easily controllable and measurable systematic error. These simulations show the phase transitions not ending at the first-order endpoint but bifurcating near there and continuing even through the expected analyticity region. The specific heat critical exponents, $\alpha$, are highly negative, explaining the lack of strong signals in the energy. Nevertheless, energy moments are seen to follow curves consistent with the predictions from the order parameters, showing these to be true thermal transitions, albeit weak ones. These results cast further doubt on Fradkin-Shenker analyticity. They also suggest one or more new yet to be explored phases in gauge-Higgs models in the bifurcated region.

hep-lat↗

arXiv2017

Two time solution to quantum measurement paradoxes

It is hypothesized that the Langevin time of stochastic quantum quantization is a physical time over which quantum fields at all values of space and coordinate time fluctuate. The average over paths becomes a time average as opposed to an ensemble average. It is further hypothesized that the Langevin time also paces the motion of particles through coordinate time and is equal to the coordinate time of the present hypersurface in the frame of the Hubble expansion. Despite having a preferred frame, special relativity continues to hold in this formulation as a dynamical symmetry due to the presumed Lorentz invariance of interactions. The measurement process becomes an integral part of the theory and is realized as a process of spontaneous symmetry breaking. The continuously fluctuating history of fields, characteristic of having two times, and the switch from ensemble averages to time averages allows for logical and straightforward explanations of many quantum measurement paradoxes. The fluctuating history also evades hidden-variable prohibitions allowing an essentially classical system to underlie quantum mechanics. These changes to the stochastic quantization paradigm makes this stochastic classical system differ somewhat from standard quantum mechanics, so, in principle, distinguishable from it.

quant-ph↗

arXiv2015

Solution to the gauge-Higgs analyticity paradox

The Fradkin-Shenker theorem proves analyticity in a region that connects Higgs to confinement regimes, precluding a phase transition. This conflicts with a simpler analyticity argument applicable to any symmetry-breaking phase transition that requires the phase diagram to be bifurcated. A flaw in the Fradkin-Shenker and related Osterwalder-Seiler proofs is found which removes this paradox. Higgs and Confinement regions are everywhere separated by a phase boundary. A new order parameter allowing this transition to be traced with Monte-Carlo simulations without gauge fixing is introduced.

hep-lat↗

arXiv2013

Direct evidence for a Coulombic phase in monopole-suppressed SU(2) lattice gauge theory

Further evidence is presented for the existence of a non-confining phase at weak coupling in SU(2) lattice gauge theory. Using Monte Carlo simulations with the standard Wilson action, gauge-invariant SO(3)-Z2 monopoles, which are strong-coupling lattice artifacts, have been seen to undergo a percolation transition exactly at the phase transition previously seen using Coulomb-gauge methods, with an infinite lattice critical point near $β= 3.2$. The theory with both Z2 vortices and monopoles and SO(3)-Z2 monopoles eliminated is simulated in the strong coupling ($β= 0$) limit on lattices up to $60^4$. Here, as in the high-$β$ phase of the Wilson action theory, finite size scaling shows it spontaneously breaks the remnant symmetry left over after Coulomb gauge fixing. Such a symmetry breaking precludes the potential from having a linear term. The monopole restriction appears to prevent the transition to a confining phase at any $β$. Direct measurement of the instantaneous Coulomb potential shows a Coulombic form with moderately running coupling possibly approaching an infrared fixed point of $α\sim 1.4$. The Coulomb potential is measured to 50 lattice spacings and 2 fm. A short-distance fit to the 2-loop perturbative potential is used to set the scale. High precision at such long distances is made possible through the use of open boundary conditions, which was previously found to cut random and systematic errors of the Coulomb gauge fixing procedure dramatically. The Coulomb potential agrees with the gauge-invariant interquark potential measured with smeared Wilson loops on periodic lattices as far as the latter can be practically measured with similar statistics data.

hep-lat↗

arXiv2013

Quantum Mechanics, Quantum Gravity, and Approximate Lorentz Invariance from a Classical Phase-Boundary Universe

A classical dynamical system in a four-dimensional Euclidean space with universal time is considered. The space is hypothesized to be originally occupied by a uniform substance, pictured as a liquid, which at some time became supercooled. Our universe began as a nucleation event initiating a liquid to solid transition. The universe we inhabit and are directly aware of consists of only the three-dimensional expanding phase boundary - a crystalline surface. Random energy transfers to the boundary from thermal fluctuations in the adjacent bulk phases are interpreted by us as quantum fluctuations, and give a physical realization to the stochastic quantization technique. Fermionic matter is modeled as screw dislocations; gauge bosons as surface acoustic waves. Minkowski space emerges dynamically through redefining local time to be proportional to the spatial coordinate perpendicular to the boundary. Lorentz invariance is only approximate, and the photon spectrum (now a phonon spectrum) has a maximum energy. Other features include a geometrical quantum gravitational theory based on elasticity theory, and a simple explanation of the quantum measurement process as a spontaneous symmetry breaking. Present, past and future are physically distinct regions, the present being a unique surface where our universe is being continually constructed.

hep-th↗

arXiv2013

Toward a proof of long range order in 4-d SU(N) lattice gauge theory

An extended version of 4-d SU(2) lattice gauge theory is considered in which different inverse coupling parameters are used, $β_H=4/g_{H}^2$ for plaquettes which are purely spacelike, and $β_V$ for those which involve the Euclidean timelike direction. It is shown that when $β_H = \infty$ the partition function becomes, in the Coulomb Gauge, exactly that of a set of non-interacting 3-d O(4) classical Heisenberg models. Long range order at low temperatures (weak coupling) has been rigorously proven for this model. It is shown that the correlation function demonstrating spontaneous magnetization in the ferromagnetic phase is a continuous function of $g_H$ at $g_H =0$ and therefore that the spontaneously broken phase enters the ($β_H$, $β_V$) phase plane (no step discontinuity at the edge). Once the phase transition line has entered, it can only exit at another identified edge, which requires the SU(2) gauge theory within also to have a phase transition at finite $β$. A phase exhibiting spontaneous breaking of the remnant symmetry left after Coulomb gauge fixing, the relevant symmetry here, is non-confining. Easy extension to the SU(N) case implies that the continuum limit of zero-temperature 4-d SU(N) lattice gauge theories is not confining, in other words gluons by themselves do not produce a confinement.

hep-lat↗

arXiv2011

Connecting phase transitions between the 3-d O(4) Heisenberg model and 4-d SU(2) lattice gauge theory

SU(2) lattice gauge theory is extended to a larger coupling space where the coupling parameter for horizontal (spacelike) plaquettes, $β_H$, differs from that for vertical (Euclidean timelike) plaquettes, $β_V$. When $β_H \rightarrow \infty$ the system, when in Coulomb Gauge, splits into multiple independent 3-d O(4) Heisenberg models on spacelike hyperlayers. Through consideration of the robustness of the Heisenberg model phase transition to small perturbations, and illustrated by Monte Carlo simulations, it is shown that the ferromagnetic phase transition in this model persists for $β_H < \infty$. Once it has entered the phase-plane it must continue to another edge due to its symmetry-breaking nature, and therefore must necessarily cross the $β_V = β_H$ line at a finite value. Indeed, a higher-order SU(2) phase transition is found at $β= 3.18 \pm 0.08$, from a finite-size scaling analysis of the Coulomb gauge magnetization from Monte Carlo simulations, which also yields critical exponents. An important technical breakthrough is the use of open boundary conditions, which is shown to reduce systematic and random errors of the overrelaxation gauge-fixing algorithm by a factor of several hundred. The string tension and specific heat are also shown to be consistent with finite-order scaling about this critical point using the same critical exponents.

hep-lat↗

arXiv2010

Is SU(2) lattice gauge theory a spin glass?

A new order parameter is constructed for SU(2) lattice gauge theory in the context of the two-real-replica method normally used for spin glasses. The order parameter is sensitive to a global Z2 subgroup of the gauge symmetry which is seen to break spontaneously at $β= 4/g^2 = 1.96\pm 0.01$. No gauge fixing is required. Finite size scaling is consistent with a high-order paramagnet to spin glass transition with a critical exponent $ν= 0.99 \pm 0.13$. The existence of this transition suggests a second transition from spin glass to ferromagnet should exist at higher $β$.

hep-lat↗

arXiv2006

Exploring Residual Gauge Symmetry Breaking

Simulations of pure-gauge SU(2) lattice gauge theory are performed in the minimal Coulomb gauge. This leaves a residual or remnant gauge symmetry still active which is global in three directions but still local in one. Using averaged fourth-dimension pointing links as a spin-like order parameter, the remnant symmetry appears to undergo spontaneous symmetry breaking at around $β= 2.6$. Both the Binder cumulant and the magnetization itself exhibit crossings in this region using lattices up to $20^4$, and a susceptibility peak is also observed. Finite size scaling indicates a weak first-order transition. The symmetry breaking is also observed to take place in the fundamental-adjoint plane, and is coincident with the strong first-order transition that exists there at large $β_{\rm{adjoint}}$. This provides confirmation that this phase transition is a symmetry-breaking transition. A well-known theorem concerning the instantaneous Coulomb potential has previously proven that a transition where such a Coulomb-gauge remnant symmetry breaks is necessarily deconfining.

hep-lat↗

arXiv2006

Spontaneous breaking of remnant gauge symmetries in zero-temperature SU(2) lattice gauge theory

The 4-d SU(2) lattice gauge theory is simulated in the minimal Coulomb gauge which aims to maximize the traces of all links in three directions. Fourth-direction links are interpreted as spins in a Heisenberg-like model with varying interactions. These spins magnetize in 3-d hyperlayers at weak coupling, breaking a remnant gauge symmetry, as well as the Polyakov-loop symmetry. They demagnetize at a phase transition around $β= 2.5$ on the infinite lattice, as determined by Binder cumulant crossings. Because $N$ symmetries are breaking on an $N^4$ lattice, the transition is unusually broad, encompassing most of the crossover region on typical lattices.

hep-lat↗

arXiv2005

Spontaneous breaking of residual gauge symmetries on the lattice

Lattice gauge theories are considered with a partial axial gauge fixing along one direction only. This leaves a residual gauge symmetry that is still local in three directions but now global in one. It is found that this $N^{d-1}$ fold symmetry (on an $N^d$ lattice) breaks spontaneously at weak coupling with the gauge field elements on links averaged over 1-d chains along the gauge-fixing direction as order parameters. This phase transition is observed with Monte-Carlo simulations for both 3-d Z2 and 4-d SU(2) pure gauge theories and appears to be coincident with the deconfinement transition. This work calls into question the equivalence of different gauges in certain circumstances.

hep-lat↗

arXiv2005

Reconsidering gauge-Higgs continuity

The 3-d Z(2) lattice gauge-Higgs theory is cast in a partial axial gauge leaving a residual Z(2) symmetry, global in two directions and local in one. It is shown both analytically and numerically that this symmetry breaks spontaneously in the Higgs phase and is unbroken in the confinement phase. Therefore they must be separated everywhere by a phase transition, in contradiction to a theorem by Fradkin and Shenker. It relied on a fully fixed unitary gauge, which prohibits this phase transition explicitly. Thus the unfixed gauge theory is not, in this case, equivalent to the unitary-gauge version.

hep-lat↗

arXiv2005

Critical or tricritical point in mixed-action SU(2) lattice gauge theory?

An analysis of scaling along the first-order bulk transition line in fundamental-adjoint SU(2) lattice gauge theory strongly supports the first-order endpoint being a tricritical point, and is inconsistent with it being an ordinary critical point as is usually assumed. If tricritical, the transition must continue from the endpoint further into the phase diagram as a second-order bulk transition and extend to and beyond the Wilson axis. Observations indicate that this is most likely the same transition that has been traditionally considered a finite-temperature transition.

hep-lat↗

arXiv2004

Evidence for layered symmetry breaking in SU(2) lattice gauge theory

Simulations of four-dimensional SU(2) lattice gauge theory are performed with partial axial gauge fixing trees spanning three of the four dimensions. The remaining SU(2) gauge symmetry, global in three directions and local in one, is found to break spontaneously at weak coupling, with the average fourth-dimension-pointing link in each perpendicular hyperplane as order parameter. The symmetry is restored at strong coupling. Symmetry breaking in each hyperplane appears to be independent, and occurs regardless of boundary conditions. The associated phase transition is likely coincident with the Polyakov loop transition.

hep-lat↗

arXiv1999

Monopole Loop Distribution and Confinement in SU(2) Lattice Gauge Theory

The abelian-projected monopole loop distribution is extracted from maximal abelian gauge simulations. The number of loops of a given length falls as a power of the length nearly independent of lattice size. This power increases with $β=4/g^2$, reaching five around $β=2.85$, beyond which loops any finite fraction of the lattice size vanish in the infinite lattice limit, suggesting the continuum theory lacks confinement.

hep-lat↗

arXiv1998

Universe as a Phase Boundary in a Four-Dimensional Euclidean Space

It is proposed that space is a four-dimensional Euclidean space with universal time. Originally this space was filled with a uniform substance, pictured as a liquid, which at some time became supercooled. Our universe began as a nucleation event initiating a liquid to solid transition. The universe we inhabit and are directly aware of consists of only the three-dimensional expanding phase boundary. Random energy transfers to the boundary from thermal fluctuations in the adjacent bulk phases are interpreted by us as quantum fluctuations. Fermionic matter is modeled as screw dislocations; gauge bosons as phonons. Minkowski space emerges dynamically through redefining local time to be proportional to the spatial coordinate perpendicular to the boundary. Other features include a geometrical quantum gravitational theory, and an explanation of quantum measurement.

gr-qc↗

arXiv1998

Do Large Abelian Monopole Loops Survive the Continuum Limit?

An analysis of the monopole loop length distribution is performed in Wilson-action SU(2) lattice gauge theory. A pure power law in the inverse length is found, at least for loops of length, $l$, less than the linear lattice size $N$. This power shows a definite $β$ dependence, passing 5 around $β=2.9$, and appears to have very little finite lattice size dependence. It is shown that when this power exceeds 5, no loops any finite fraction of the lattice size will survive the infinite lattice limit. This is true for any reasonable size distribution for loops larger than N. The apparent lack of finite size dependence in this quantity would seem to indicate that abelian monopole loops large enough to cause confinement do not survive the continuum limit. Indeed they are absent for all $β> 2.9$.

hep-lat↗